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受欢迎的三角函数 >

2sin^2(4x)-2cos(4x)=1,o<= x<= 360

解答

2 sin^{2} (4 x) - 2 cos (4 x) = 1, o^{\circ} \leq x \leq 36 0^{\circ}

解答

x \in R 无解

求解步骤

2 sin^{2} (4 x) - 2 cos (4 x) = 1, o^{\circ} \leq x \leq 36 0^{\circ}

两边减去

1

2 sin^{2} (4 x) - 2 cos (4 x) - 1 = 0

使用三角恒等式改写

- 1 - 2 cos (4 x) + 2 sin^{2} (4 x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 1 - 2 cos (4 x) + 2 (1 - cos^{2} (4 x))

化简

- 1 - 2 cos (4 x) + 2 (1 - cos^{2} (4 x)) : - 2 cos^{2} (4 x) - 2 cos (4 x) + 1

- 1 - 2 cos (4 x) + 2 (1 - cos^{2} (4 x))

乘开

2 (1 - cos^{2} (4 x)) : 2 - 2 cos^{2} (4 x)

2 (1 - cos^{2} (4 x))

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (4 x)

= 2 \cdot 1 - 2 cos^{2} (4 x)

数字相乘:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (4 x)

= - 1 - 2 cos (4 x) + 2 - 2 cos^{2} (4 x)

化简

- 1 - 2 cos (4 x) + 2 - 2 cos^{2} (4 x) : - 2 cos^{2} (4 x) - 2 cos (4 x) + 1

- 1 - 2 cos (4 x) + 2 - 2 cos^{2} (4 x)

对同类项分组

= - 2 cos (4 x) - 2 cos^{2} (4 x) - 1 + 2

数字相加/相减:

- 1 + 2 = 1

= - 2 cos^{2} (4 x) - 2 cos (4 x) + 1

= - 2 cos^{2} (4 x) - 2 cos (4 x) + 1

= - 2 cos^{2} (4 x) - 2 cos (4 x) + 1

1 - 2 cos (4 x) - 2 cos^{2} (4 x) = 0

用替代法求解

1 - 2 cos (4 x) - 2 cos^{2} (4 x) = 0

令

: cos (4 x) = u

1 - 2 u - 2 u^{2} = 0

1 - 2 u - 2 u^{2} = 0 : u = - \frac{1 + 3}{2}, u = \frac{3 - 1}{2}

1 - 2 u - 2 u^{2} = 0

改写成标准形式

a x^{2} + b x + c = 0

- 2 u^{2} - 2 u + 1 = 0

使用求根公式求解

- 2 u^{2} - 2 u + 1 = 0

二次方程求根公式:

若

a = - 2, b = - 2, c = 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

(- 2)^{2} - 4 (- 2) \cdot 1 = 23

(- 2)^{2} - 4 (- 2) \cdot 1

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 2 \cdot 1

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

数字相加:

4 + 8 = 12

= 12

12

质因数分解:

2^{2} \cdot 3

12

12

除以

212 = 6 \cdot 2

= 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 23

u_{1, 2} = \frac{- ( - 2 ) \pm 2 3}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 2 3}{2 ( - 2 )}, u_{2} = \frac{- ( - 2 ) - 2 3}{2 ( - 2 )}

u = \frac{- ( - 2 ) + 2 3}{2 ( - 2 )} : - \frac{1 + 3}{2}

\frac{- ( - 2 ) + 2 3}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 + 2 3}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 + 2 3}{- 4}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 + 2 3}{4}

消掉

\frac{2 + 2 3}{4} : \frac{1 + 3}{2}

\frac{2 + 2 3}{4}

分解

2 + 23 : 2 (1 + 3)

2 + 23

改写为

= 2 \cdot 1 + 23

因式分解出通项

2

= 2 (1 + 3)

= \frac{2 ( 1 + 3 )}{4}

约分:

2

= \frac{1 + 3}{2}

= - \frac{1 + 3}{2}

u = \frac{- ( - 2 ) - 2 3}{2 ( - 2 )} : \frac{3 - 1}{2}

\frac{- ( - 2 ) - 2 3}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 - 2 3}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 - 2 3}{- 4}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

2 - 23 = - (23 - 2)

= \frac{2 3 - 2}{4}

分解

23 - 2 : 2 (3 - 1)

23 - 2

改写为

= 23 - 2 \cdot 1

因式分解出通项

2

= 2 (3 - 1)

= \frac{2 ( 3 - 1 )}{4}

约分:

2

= \frac{3 - 1}{2}

二次方程组的解是:

u = - \frac{1 + 3}{2}, u = \frac{3 - 1}{2}

u = cos (4 x)

代回

cos (4 x) = - \frac{1 + 3}{2}, cos (4 x) = \frac{3 - 1}{2}

cos (4 x) = - \frac{1 + 3}{2}, cos (4 x) = \frac{3 - 1}{2}

cos (4 x) = - \frac{1 + 3}{2}, o^{\circ} \leq x \leq 36 0^{\circ} :

无解

cos (4 x) = - \frac{1 + 3}{2}, o^{\circ} \leq x \leq 36 0^{\circ}

- 1 \leq cos (x) \leq 1

无解

cos (4 x) = \frac{3 - 1}{2}, o^{\circ} \leq x \leq 36 0^{\circ} :

无解

cos (4 x) = \frac{3 - 1}{2}, o^{\circ} \leq x \leq 36 0^{\circ}

使用反三角函数性质

cos (4 x) = \frac{3 - 1}{2}

cos (4 x) = \frac{3 - 1}{2}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

4 x = arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n, 4 x = 36 0^{\circ} - arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

4 x = arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n, 4 x = 36 0^{\circ} - arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

解

4 x = arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n : x = \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

4 x = arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

两边除以

4

4 x = arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

两边除以

4

\frac{4 x}{4} = \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4}

化简

x = \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

x = \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

解

4 x = 36 0^{\circ} - arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n : x = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

4 x = 36 0^{\circ} - arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

两边除以

4

4 x = 36 0^{\circ} - arccos (\frac{3 - 1}{2}) + 36 0^{\circ} n

两边除以

4

\frac{4 x}{4} = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4}

化简

\frac{4 x}{4} = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4} : 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4}

消掉

9 0^{\circ} : 9 0^{\circ}

9 0^{\circ}

约分:

2

= 9 0^{\circ}

= 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{36 0 ^{\circ} n}{4}

消掉

\frac{36 0 ^{\circ} n}{4} : \frac{18 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{4}

约分:

2

= \frac{18 0 ^{\circ} n}{2}

= 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

x = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

x = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

x = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

x = \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}, x = 9 0^{\circ} - \frac{arccos ( \frac{3 - 1}{2} )}{4} + \frac{18 0 ^{\circ} n}{2}

在

o^{\circ} \leq x \leq 36 0^{\circ}

范围内的解

无解

合并所有解

x \in R 无解

作图

流行的例子

sin (x) - 3 = 0

2sqrt(3)+4cos(θ)=0

23 + 4 cos (θ) = 0

2cos(x)=1,0<= x<2pi

2 cos (x) = 1, 0 \leq x < 2 π

cos (x) = - \frac{3}{2}

tan^2(x)+2tan(x)-3=0

tan^{2} (x) + 2 tan (x) - 3 = 0